Qt Quick Designer uses an external process called qmlpuppet to render components.
By default a qmlpuppet shipped together with Qt Creator is used. This one does not support
QtDesktop components.

The solution is to build share\qtcreator\qml\qmlpuppet in the Qt Creator source directory with a Qt version that supports the required components (in this case the QtDesktop components).

Building qmlpuppet will install an executable for rendering components in the /bin directory of the Qt building it. Qt Quick Designer checks for the Qt of the current project if a qmlpuppet is in the /bin directory. If yes it takes the qmlpuppet provided by the Qt version instead of the one provided by Qt Creator itself. This is a generic solution to be able to render any component available in a specific Qt.

For the ‘invalid property name shortcut’ error message I have no immediate solution.